To determine how far the stake needs to be from the middle of the tent, we can model the situation as a right triangle.

• The height from the top of the tent to the ground (the vertical leg) is 4 feet.
• The length of the rope (the hypotenuse) is 8 feet.
• The distance from the middle of the tent to the stake (the horizontal leg) is what we need to find.

Let this distance be $x$. We can use the Pythagorean theorem:

$a^2 + b^2 = c^2$

Here, $a = 4$ feet, $c = 8$ feet, and $b = x$. Substituting into the equation:

$4^2 + x^2 = 8^2$

$16 + x^2 = 64$

Now, solve for $x^2$:

$x^2 = 64 - 16$

$x^2 = 48$

Taking the square root of both sides:

$x = \sqrt{48} \approx 6.9 \text{ feet}$

So, the stake needs to be about 6.9 feet from the middle of the tent.

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. What if the rope length was 10 feet instead of 8 feet? How would that change the distance to the stake?
2. How would you determine the distance if the tent height was 6 feet instead of 4 feet?
3. Can you find the length of the rope needed if the stake must be 5 feet away from the tent?
4. How would the problem change if the stake was placed at an angle other than perpendicular to the tent?
5. How could trigonometry be used to solve this problem instead of the Pythagorean theorem?

Tip: When dealing with right triangles, always check if the given information fits the Pythagorean theorem before solving.